## 37.63 Ind-quasi-affine morphisms

A bit of theory to be used later.

Definition 37.63.1. A scheme $X$ is ind-quasi-affine if every quasi-compact open of $X$ is quasi-affine. Similarly, a morphism of schemes $X \to Y$ is ind-quasi-affine if $f^{-1}(V)$ is ind-quasi-affine for each affine open $V$ in $Y$.

An example of an ind-quasi-affine scheme is an open of an affine scheme. If $X = \bigcup _{i \in I} U_ i$ is a union of quasi-affine opens such that any two $U_ i$ are contained in a third, then $X$ is ind-quasi-affine. An ind-quasi-affine scheme $X$ is separated because any two affine opens $U, V$ are contained in a separated open subscheme of $X$, namely $U \cup V$. Similarly an ind-quasi-affine morphism is separated.

Lemma 37.63.2. For a morphism of schemes $f : X \to Y$, the following are equivalent:

1. $f$ is ind-quasi-affine,

2. for every affine open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

3. for some cover $\{ V_ j \} _{j \in J}$ of $Y$ by quasi-compact and quasi-separated open subschemes $V_ j \subset Y$, every $j \in J$, and every quasi-compact open subscheme $U \subset f^{-1}(V_ j)$, the induced morphism $U \to V_ j$ is quasi-affine.

4. for every quasi-compact and quasi-separated open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base.

Proof. The equivalence (1) $\Leftrightarrow$ (2) follows from the definitions and Morphisms, Lemma 29.13.3. For (2) $\Rightarrow$ (4), let $U$ and $V$ be as in (4). By Schemes, Lemma 26.21.14, the induced morphism $U \to V$ is quasi-compact. Thus, for every affine open $V' \subset V$, the fiber product $V' \times _ V U$ is quasi-compact, so, by (2), the induced map $V' \times _ V U \to V'$ is quasi-affine. Thus, $U \to V$ is also quasi-affine by Morphisms, Lemma 29.13.3. This argument also gives (3) $\Rightarrow$ (4): indeed, keeping the same notation, those affine opens $V' \subset V$ that lie in one of the $V_ j$ cover $V$, so one needs to argue that the quasi-compact map $V' \times _ V U \to V'$ is quasi-affine. However, by (3), the composition $V' \times _ V U \to V' \to V_ j$ is quasi-affine and, by Schemes, Lemma 26.21.13, the map $V' \to V_ j$ is quasi-separated. Thus, $V' \times _ V U \to V'$ is quasi-affine by Morphisms, Lemma 29.13.8. The final implications (4) $\Rightarrow$ (2) and (4) $\Rightarrow$ (3) are evident. $\square$

Lemma 37.63.3. The property of being an ind-quasi-affine morphism is stable under composition.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be ind-quasi-affine morphisms. Let $V \subset Z$ and $U \subset f^{-1}(g^{-1}(V))$ be quasi-compact opens such that $V$ is also quasi-separated. The image $f(U)$ is a quasi-compact subset of $g^{-1}(V)$, so it is contained in some quasi-compact open $W \subset g^{-1}(V)$ (a union of finitely many affines). We obtain a factorization $U \to W \to V$. The map $W \to V$ is quasi-affine by Lemma 37.63.2, so, in particular, $W$ is quasi-separated. Then, by Lemma 37.63.2 again, $U \to W$ is quasi-affine as well. Consequently, by Morphisms, Lemma 29.13.4, the composition $U \to V$ is also quasi-affine, and it remains to apply Lemma 37.63.2 once more. $\square$

Lemma 37.63.4. Any quasi-affine morphism is ind-quasi-affine. Any immersion is ind-quasi-affine.

Proof. The first assertion is immediate from the definitions. In particular, affine morphisms, such as closed immersions, are ind-quasi-affine. Thus, by Lemma 37.63.3, it remains to show that an open immersion is ind-quasi-affine. This, however, is immediate from the definitions. $\square$

Lemma 37.63.5. If $f : X \to Y$ and $g : Y \to Z$ are morphisms of schemes such that $g \circ f$ is ind-quasi-affine, then $f$ is ind-quasi-affine.

Proof. By Lemma 37.63.2, we may work Zariski locally on $Z$ and then on $Y$, so we lose no generality by assuming that $Z$, and then also $Y$, is affine. Then any quasi-compact open of $X$ is quasi-affine, so Lemma 37.63.2 gives the claim. $\square$

Lemma 37.63.6. The property of being ind-quasi-affine is stable under base change.

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. For checking that every base change of $f$ is ind-quasi-affine, by Lemma 37.63.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Furthermore, we may also assume that in the base change morphism $Z \to Y$ the scheme $Z$ is affine, too. The base change $X \times _ Y Z \to X$ is an affine morphism, so, by Lemmas 37.63.3 and 37.63.4, the map $X \times _ Y Z \to Y$ is ind-quasi-affine. Then, by Lemma 37.63.5, the base change $X \times _ Y Z \to Z$ is ind-quasi-affine, as desired. $\square$

Lemma 37.63.7. The property of being ind-quasi-affine is fpqc local on the base.

Proof. The stability of ind-quasi-affineness under base change supplied by Lemma 37.63.6 gives one direction. For the other, let $f : X \to Y$ be a morphism of schemes and let $\{ g_ i : Y_ i \to Y\}$ be an fpqc covering such that the base change $f_ i : X_ i \to Y_ i$ is ind-quasi-affine for all $i$. We need to show $f$ is ind-quasi-affine.

By Lemma 37.63.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Then we use stability under base change ensured by Lemma 37.63.6 to refine the cover and assume that it is given by a single affine, faithfully flat morphism $g : Y' \to Y$. For any quasi-compact open $U \subset X$, its $Y'$-base change $U \times _ Y Y' \subset X \times _ Y Y'$ is also quasi-compact. It remains to observe that, by Descent, Lemma 35.22.20, the map $U \to Y$ is quasi-affine if and only if so is $U \times _ Y Y' \to Y'$. $\square$

Lemma 37.63.8. A separated locally quasi-finite morphism of schemes is ind-quasi-affine.

Proof. Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes. Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact open. We have to show $U$ is quasi-affine. Since $U \to V$ is a separated quasi-finite morphism of schemes, this follows from Zariski's Main Theorem. See Lemma 37.42.2. $\square$

Comment #1785 by Laurent Moret-Bailly on

Examples after the definition: an open of a quasi-projective scheme is not ind-quasi-affine in general! On the other hand, (an open of) a disjoint sum of quasi-affine schemes is, or more generally (an open of) a scheme with a filtered open covering by quasiaffines. (Any interesting examples of the latter?)

Typo in proof of 36.49.2: $\bigcup V_i$.

Comment #1788 by Laurent Moret-Bailly on

Examples after the definition: an open of a quasi-projective scheme (e.g. $\mathbb{P}^1$) is not ind-quasi-affine in general! One could mention disjoint sums of quasi-affine schemes (but this is not a "new" example since $\coprod_i\mathrm{Spec}\,(R_i)$ is open in $\mathrm{Spec}\,(\prod_i R_i)$). More generally (an open of) a scheme with a filtered open covering by quasiaffines is ind-quasi-affine. Any interesting examples of the latter?

Typo in proof of 36.49.2: $\bigcup V_i$.

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